# The network complexity and the Turing machine complexity of finite functions

## Summary

Let L(f) be the network complexity of a Boolean function L(f). For any n-ary Boolean function L(f) let $$TC(f) = min\{ T_p^{\bar A} (n){\text{ (}}\parallel p\parallel + 1gS_p^{\bar A} {\text{(}}n{\text{):}}res_p^{\bar A} {\text{(}}n{\text{) = }}f\}$$. Hereby p ranges over all relative Turing programs and Ā ranges over all oracles such that given the oracle Ā, the restriction of p to inputs of length n is a program for L(f). ∥p∥ is the number of instructions of p. T p Ā(n) is the time bound and S p Āof the program p relative to the oracle Ā on inputs of length n. Our main results are (1) L(f) ≦ O(TC(L(f))), (2) TC(f) ≦ O(L(f) 2 2+ɛ) for every ɛ ⋙ O.

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The results of this paper have been reported in a main lecture at the 1975 annual meeting of GAMM, April 2–5, Göttingen

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Schnorr, C.P. The network complexity and the Turing machine complexity of finite functions. Acta Informatica 7, 95–107 (1976). https://doi.org/10.1007/BF00265223